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Vol. 15 (2010), Paper no. 3, pages 75–95.
Journal URL
http://www.math.washington.edu/~ejpecp/
On the Shuffling Algorithm for Domino Tilings∗
Eric Nordenstam
Institutionen för Matematik
Swedish Royal Institute of Technology (KTH)
100 44 Stockholm, Sweden
eno@math.kth.se
Abstract
We study the dynamics of a certain discrete model of interacting interlaced particles that comes
from the so called shuffling algorithm for sampling a random tiling of an Aztec diamond. It
turns out that the transition probabilities have a particularly convenient determinantal form. An
analogous formula in a continuous setting has recently been obtained by Jon Warren studying
certain model of interlacing Brownian motions which can be used to construct Dyson’s nonintersecting Brownian motion.
We conjecture that Warren’s model can be recovered as a scaling limit of our discrete model and
prove some partial results in this direction. As an application to one of these results we use it
to rederive the known result that random tilings of an Aztec diamond, suitably rescaled near a
turning point, converge to the GUE minor process.
Key words: random tilings; Brownian motion; random matrices.
AMS 2000 Subject Classification: Primary 60C05; 60G50.
Submitted to EJP on February 19, 2008, final version accepted February 23, 2009.
∗
Supported by grant KAW 2005.0098 from the Knut and Alice Wallenberg Foundation and by the Göran Gustafsson
foundation (KVA)
75
1
Introduction
There has been a lot of work in recent years connecting tilings of various planar regions with random
matrices. One particular model that has been intensely studied is domino tilings of a so called Aztec
diamond. One way of analysing that model, [Joh05; Joh01; JN06], is to define a particle process
corresponding to the tilings so that the uniform measure on all tilings induces some measure on this
particle process.
In this article we study the so called shuffling algorithm, described in [EKLP92; Pro03], which in
various variants can be used either to count or to enumerate all tilings of the Aztec diamond or to
sample a random such tiling.
The sampling of a random tiling by this method is an iterative process. Starting with a tiling of an
order n − 1 Aztec diamond, a certain procedure is performed, producing a random tiling of order n.
This procedure is usually described in terms of the dominoes which should be moved and created
according to a certain procedure. We will instead look at this algorithm as a certain dynamics on
the particle process mentioned above.
The detailed dynamics of the particle process will be presented in section 2 and how it is obtained
from the traditional formulation of the shuffling algorithm is presented in section 3. For now, consider a process X (t) = (X 1 (t), . . . , X m (t)) for t = 0, 1, 2, . . . , where X n (t) = (X 1n (t), . . . , X nn (t)) ∈ Zn .
The quantity X in (t) represents the position of the i:th particle on line n after t − n steps of the shuffling algorithm have been performed. At each time a certain interlacing condition (3) is maintained.
(The reason for the t − n is technical convenience.)
Denote by X N (t ) = (X N ,1 (t ), . . . , X N ,m (t )), for t ∈ [0, ∞), a version of X (t) rescaled according to
N ,n
X i (t )
X in (N t ) − 12 N t
=
,
1p
N
2
t∈
1
N
N0 ,
(1)
and extended by linear interpolation to non-integer values of N t . We will prove the following in
section 7.
Theorem 1.1. For fixed n, as N → ∞, the process (X N ,n (t )) t∈[0,∞) converges weakly to a Dyson
Brownian motion with all n particles started at the origin.
The full process (X (t)) t∈N0 has remarkable similarities to, and is we believe a discretization of,
a process studied recently by Warren, [War07]. It consists of many interlaced Dyson Brownian
motions and is here briefly described in section 4. We will denote that process (X(t ))t ∈[0,∞) . We
show the following in section 7 along with the stronger statement Theorem 7.4.
Theorem 1.2. Let t ≥ 0 be fixed. Then X N (t ) converges in distribution to X(t ) as N → ∞.
The key to our asymptotic analysis of the shuffling algorithm is that the transition probabilities of
(X n (t), X n+1 (t)) t∈N0 can be written in a convenient determinantal form, see proposition 4.2. These
formulas mirror in a beautiful way formulas obtained by Warren.
As an application of our results we will use it to rederive an asymptotic result about random tilings
near the point where the arctic circle touches the edge of the diamond. This result was first stated
in [Joh05] and proved in [JN06].
76
Recall that the Gaussian Unitary Ensemble, or GUE for short, is a probability measure on m × m Her−1 − Tr H 2 /2
mitian matrices with density Zm
e
where Zm is a normalisation constant. Let H = (h rs )1≤r,s≤m
be a GUE matrix and denote its principal minors by H n = (h rs )1≤r,s≤n . Let λn = (λ1n , . . . , λnn )
be the vector of eigenvalues of H n ordered so that λni ≤ λni+1 for i = 1, . . . , n − 1. Then
Λ = (λ1 , . . . , λm ) ∈ Rm(m+1)/2 is in [JN06] called the GUE minor process.
Corollary 1.3 (Theorem 1.5 in [JN06]). X N (1) → Λ in distribution as N → ∞.
Proof. Warren in [War07] shows that Λ has the same distribution as X(1).
To put this in perspective, let us note that a similar result for lozenge tilings is known from Okounkov
and Reshetikhin [OR06]. They discuss the fact that, for quite general regions, that close to a so
called turning point the GUE minor process can be obtained in a limit. A turning point is, just as in
our situation, where the disordered region is tangent to the domain boundary.
After this work appeared as a preprint, Borodin & Ferrari published [BF08] treating a class of models
where the present particle model appears as a special case. They however approach the problem
from a different angle and study different scaling limits. The transition probabilities they give are
on a different form from the ones we give in section 2. It is not obvious how to algebraically relate
these two expressions for the same thing, and it would be interesting to understand this.
Borodin & Gorin in [BG08] do a similar analysis to the one in this article in the case of tilings of a
hexagon with rhombuses. Their construction also fits into the general framework of [BF08].
Acknowledgements: The author would like to thank his supervisor Kurt Johansson for many useful
discussions.
2
The Aztec Diamond Particle Process
We will here content ourselves with stating the rules of the particle dynamics that we will study. The
reader will in section 3 find a summary of the traditional formulation of the shuffling algorithm and
how it relates to the formulas below.
Consider the process (X (t)) = (X 1 (t), . . . , X m (t)) for t = 0, 1, 2, . . . , where X n (t) =
(X 1n (t), . . . , X nn (t)) ∈ Zn . It satisfies the initial condition
X n (0) = x̄ n
(2)
for n = 0, . . . , m where x̄ in = i for 1 ≤ i ≤ n. At each time t ∈ N0 the process fulfils the interlacing
condition
n
X in (t) ≤ X in−1 (t) < X i+1
(t), for 1 ≤ i < n,
(3)
77
and evolves in time according to
X 11 (t) = X 11 (t − 1) + β11 (t),
X 1n (t) = X 1n (t − 1) + β1n (t)
− 1{X 1n (t − 1) + β1n (t) = X 1n−1 (t) + 1}, for n ≥ 2,
X nn (t) = X nn (t − 1) + βnn (t)
n−1
+ 1{X nn (t − 1) + βnn (t) = X n−1
(t)},
j
j
for n ≥ 2,
(4)
j
X i (t) = X i (t − 1) + βi (t)
j
j
j−1
j
j
j−1
− 1{X i (t − 1) + βi (t) = X i
(t) + 1}
+ 1{X i (t − 1) + βi (t) = X i−1 (t)},
for n ≥ 3 and 1 < i < n,
and t ∈ N0 . All the βin (t) for 1 ≤ i ≤ n and t ∈ N are i.i.d. unbiased coin tosses, satisfying
P[β11 (1) = 0] = P[β11 (1) = 1] = 21 .
One way to think about this is that at each time t, this is a set of particles on m lines. The n:th line
has n particles on it at positions X 1n < · · · < X nn . At each time step each of these particles either stays
or jumps one unit step forward independent of all others except that the particles on line n can push
or block the particles on line n + 1 to enforce the interlacing condition (3). The lines are updated in
sequence starting with line 1 and ending in line m. On each line the order of update of the particles
is irrelevant.
As mentioned we can write down transition probabilities for this process on a particularly convenient
determinantal form. Define the delta function δi : Z → R such that δi (x) = 1 if i = x and δi (x) = 0
otherwise. Let us first introduce some notation.
(φ ∗ ψ)(x) =
X
φ(s)ψ(t)
(Convolution product)
s+t=x
φ (0) = δ0
φ (n) = φ (n−1) ∗ φ
for n = 1, 2, . . .
∆φ = (δ0 − δ1 ) ∗ φ
x
X
∆−1 φ(x) =
φ( y) = (δ0 + δ1 + . . . ) ∗ φ
(Backward difference)
y=−∞
These convolutions have the following probabilistic meaning. Let X 1 , X 2 , . . . , be a sequence of
i.i.d. random variables with probability distribution φ. Then Sn = X 1 + · · · + X n has probability
distribution φ (n) . Backward difference operator ∆ and the summation operator ∆−1 have no such
simple probabilistic meaning. In the rest of this article φ = 12 (δ0 + δ1 ) which is the case of a
Bernoulli random walk.
Let W n+1,n = {(x, y) ∈ Zn+1 × Zn : x 1 ≤ y1 < x 2 ≤ · · · ≤ yn < x n+1 }. For (x, y), (x 0 , y 0 ) ∈ W n+1,n
and t = N0 , define
–
™
A t (x, x 0 ) B t (x, y 0 )
n
0
0
q t ((x, y), (x , y )) = det
(5)
C t ( y, x 0 ) D t ( y, y 0 )
78
where
• A t (x, x 0 ) is an (n + 1) × (n + 1)-matrix where element (i, j) is φ (t) (x i0 − x j ),
• B t (x, y 0 ) is an (n + 1) × (n)-matrix where element (i, j) is ∆−1 φ (t) ( yi0 − x j ) − 1{ j ≥ i},
• C t ( y, x 0 ) is an n × (n + 1)-matrix where element (i, j) is ∆φ (t) (x i0 − y j ) and
• D t ( y, y 0 ) is an n × n-matrix where element (i, j) is φ (t) ( yi0 − y j ).
Note that the expression ∆−1 φ (t) is taken to mean ∆−1 (φ (t) ), not (∆−1 φ)(t) . As a side note,
and this will be useful in later sections, convolution is a commutative operation. So for example
∆−1 (φ (t) ) = (∆−1 φ) ∗ φ (t−1) for t ∈ N.
Let W n = {x ∈ Zn : x 1 < x 2 < · · · < x n } and for x ∈ W n let the Vandermonde determinant be
Y
hn (x) =
(x j − x i ).
(6)
1≤i< j≤n
Theorem 2.1. The transition probabilities of (X n (t), X n+1 (t)) t∈N0 from the process (X (t) t∈N0 above
are
hn ( y 0 ) n
q ((x, y), (x 0 , y 0 )),
(7)
q tn,+ ((x, y), (x 0 , y 0 )) :=
hn ( y) t
that is
P[(X n+1 (s + t), X n (s + t)) = (x 0 , y 0 )|(X n+1 (s), X n (s)) = (x, y)]
n,+
= qt
((x, y), (x 0 , y 0 )). (8)
A proof is given in section 6. It is a very straightforward computation to integrate out the x component in expression (7). We find that the transition probabilities of (X n ) t∈N0 from the process (X ) t∈N0
are
hn ( y 0 ) n
p tn,+ ( y, y 0 ) :=
p ( y, y 0 )
(9)
hn ( y) t
where p tn ( y, y 0 ) := D t ( y, y 0 ) given above. We recognise this transition probability as a KarlinMacGregor type determinant with a Doob h-conditioning. This leads to the following important
observation.
Corollary 2.2. The component (X n (t)) t∈N0 of (X (t)) t∈N0 is the positions of n walkers started at x̄ n ,
taking steps with distribution φ and conditioned never to intersect.
This fits nicely with theorem 1.1. The process (X n (t)) t∈N0 is a discrete Dyson Brownian motion of n
particles and its limit under suitable rescaling is Brownian motions conditioned never to intersect,
which is exactly what (X n (t ))t ≥0 from Warren’s process (X(t ))t ≥0 is.
79
3
The Shuffling algorithm
We will now relate some well known facts about sampling random tilings of an Aztec diamond
before showing how to get the particle dynamics in section 2.
The Aztec diamond of order n, denoted An , is an area in the plane that is the union of those lattice
squares [a, a + 1] × [b, b + 1] ⊂ R2 for a, b ∈ Z that are entirely contained in the square {(x, y) ∈
R2 : |x| + | y| ≤ n + 1}. An can be tiled in 2n(n+1)/2 ways by dominoes that are of size 2 × 1. We will
be interested picking a random tiling. By random tiling in this article we will always mean that all
possible tilings are given the same probability.
A key ingredient of almost all results concerning tilings of the Aztec diamond is the realization that
one can distinguish four kinds of dominoes present in a typical tiling. The obvious distinction to the
casual observer is the difference between horizontal and vertical dominoes. These can be subdivided
further. Colour the underlying lattice squares black and white according to a checkerboard fashion
in such a way that the left square on the top line is black. Let a horizontal domino be of type N or
north if its leftmost square is black, and of type S or south otherwise. Likewise let a vertical domino
be of type W or west if its topmost square is black and type E or east otherwise. In figures 1 and 2
the S and E type dominoes have been shaded for convenience.
One way of sampling from this measure is the so called shuffling algorithm, first described
in [EKLP92], and very nicely explained and generalised in [Pro03]. It is an iterative procedure
that produces a random tiling of An+1 given a random tiling of An and some number of coin-tosses.
One starts with the empty tiling on A0 and one repeats this process until one has a tiling of the
desired size. It is a theorem that this procedure gives all tilings with equal probability, provided that
the coin-tosses made along the way were fair.
The algorithm works in three stages. Start with a tiling of An .
Destruction All 2 × 2 blocks consisting of an S-domino directly above an N-domino are removed.
Likewise all 2 × 2 blocks of consisting of an E-domino directly to left of a W-domino are
removed.
Shuffling All N, S, E and W-dominoes respectively move one unit length up, down, right and left
respectively.
Creation The result is a tiling of a subset of An+1 . The empty parts can be covered in a unique way
by 2×2 squares. Toss a coin to fill these with two horizontal or two vertical dominoes with
equal probability.
Figure 1 illustrates the process. In the leftmost column there are tilings of successively larger diamonds. From column one to column two, the destruction step is carried out. From there to the
third column, shuffling is performed. These figures contain several dots which will concern us later
in this exposition. The creation step of the algorithm applied to a diamond in the last column gives
(with positive probability) the diamond in the first column on the next row.
To study more detailed properties of random tilings it is useful to introduce a coordinate system
suited to the setting and a particle process such that the possible tilings correspond to particle
configurations.
In the left picture in figure 2, the S and E type dominoes are shaded and a coordinate system is
imposed on the tiling. For each tile there is exactly one of the x lines and exactly one of the y lines
80
Figure 1: The shuffling procedure. S- and E-type dominoes are shaded.
81
that passes through its interior. Indeed we can uniquely specify the location of a tile by giving its
coordinates (x, y) and type (N, S, E or W). One can see that along the line y = n there are exactly
n shaded tiles, for y = 1, . . . , 8 where 8 is the order of the diamond. The generalisation of that
statement is true for tilings of An for any n. We shall call the occurrence of a shaded tile a particle.
The right picture in figure 2 is the same tiling but with dots marking the particles.
j
Just to fix some notation, let x i be the x-coordinate of the i:th particle along the line y = j. It is
clear from the definitions that these satisfy an interlacing condition,
j
j−1
xi ≤ xi
j
≤ x i+1 .
(10)
We will now see how the shuffling algorithm described above acts on these particles.
It turns out that the positions of the particles is uniquely determined before the creation stage of
the last iteration of the shuffling algorithm, and we have marked these with dots in the last column
in figure 1. As can be seen in that figure, running the shuffling algorithm to produce tilings of
successively larger Aztec diamonds imposes certain dynamics on these particles. That is the central
object of study in this article.
Let us first consider the trajectory of x 11 . As can easily be seen in figure 1, on the y = 1 line there
are always a number of W-dominoes, then the particle, then a number of N-dominoes. Depending
on whether the creation stage of the algorithm fills the empty space in between these with a pair of
horizontal or vertical dominoes, either the particle stays or its x-coordinate will increase by one in
the next step. Thus the first particle performs the simple random walk
x 11 (t) = x 11 (t − 1) + γ11 (t).
j
(11)
j
were γij (t) are independent coin tosses, i.e. P[γi (t) = 1] = P[γi (t) = 0] = 21 , for t, j = 1, . . . and
0 ≤ i ≤ j.
Consider now the particles on row y = 2. For x 12 , while x 12 (t) < x 11 (t) it performs a random walk
independently of x 11 , at each time either staying or adding one with equal probability. However,
when there is equality, x 12 (t) = x 11 (t), then the particle must be represented by a vertical (S) tile.
Thus it does not contribute to growth of the west polar region, thus the particle will remain fixed.
In order to represent this as a formula, we subtract one if the particle attempts to jump past x 11 .
x 12 (t) = x 12 (t − 1) + γ21 (t) − 1{x 12 (t − 1) + γ21 (t) = x 11 (t − 1) + 1}
(12)
Symmetry completes our analysis of this row with the relation
x 22 (t) = x 22 (t − 1) + γ22 (t) + 1{x 22 (t − 1) + γ22 (t) = x 11 (t − 1)}.
(13)
For the third row, our previous analysis applies to the first and last particle.
x 13 (t) = x 13 (t − 1) + γ31 (t) − 1{x 13 (t − 1) + γ31 (t) = x 12 (t − 1) + 1}
(14)
x 33 (t)
(15)
=
x 33 (t
− 1) + γ33 (t) + 1{x 33 (t
− 1) + γ33 (t)
=
x 22 (t
− 1)}
On y = 3 between x 12 and x 22 there must be first a sequence of zero or more E dominoes, then x 23 ,
then a sequence of zero or more N dominoes. While x 23 is in the interior of this area it performs the
customary random walk. It must interact with x 12 and x 22 in the same way as we have seen other
particles interacting above.
82
So
x 23 (t) = x 23 (t − 1) + γ32 (t) − 1{x 23 (t − 1) + γ32 (t) = x 22 (t − 1) + 1}
+ 1{x 23 (t − 1) + γ32 (t) = x 12 (t − 1)}.
(16)
The same pattern repeats itself evermore.
j
j
j
j
j
j−1
x 1 (t) = x 1 (t − 1) + γ1 (t) − 1{x 1 (t − 1) + γ1 (t) = x 1 (t − 1) + 1}
(17)
j
x j (t)
j
x i (t)
(18)
=
=
j
x j (t
j
x i (t
j
j
j
j−1
− 1) + γ j (t) + 1{x j (t − 1) + γ j (t) = x j−1 (t − 1)}
j
j
j
j−1
− 1) + γi (t) − 1{x i (t − 1) + γi (t) = x j (t − 1) + 1}
j
j
j−1
+ 1{x i (t − 1) + γi (t) = x j−1 (t − 1)}.
(19)
(20)
j
with initial conditions x i ( j) = i for j = 2, . . . and 1 ≤ i ≤ j.
x=8
x=7
x=6
x=5
x=4
x=3
x=2
x=1
y=1
x=8
y=2
x=7
y=3
x=6
y=4
x=5
y=5
x=4
y=6
x=3
y=7
x=2
y=8 x=1
y=1
y=2
y=3
y=4
y=5
y=6
y=7
y=8
Figure 2: Same diamond
In order to analyse this process it is suitable to perform a change of variables,
j
j
X i (t) = x i (t − j),
(21)
which gives the equations given in section 2.
4
Interlacing Brownian motions
We will now digress a bit and summarise Warren’s work in [War07], so as to fix notation and to
emphasise the similarities between his continuous process and our discrete process. The reader
is referred to that reference for more details of the construction. Consider an Rn+1 × Rn -valued
stochastic process (Q(t)) t≥0 = (X (t), Y (t)) t≥0 satisfying an interlacing condition
X 1 (t ) ≤ Y1 (t ) ≤ X 2 (t ) ≤ · · · ≤ Yn (t ) ≤ X n+1 (t ),
(22)
and equations
Yi (t ) = yi + βi (t ∧ τ),
X i (t ) = yi + γi (t ∧ τ) +
L i− (t
∧ τ) −
L i+ (t
83
∧ τ)
for i = 1, . . . , n,
(23)
for i = 1, . . . , n + 1,
(24)
where
βi for i = 1, . . . , n and γi for i = 1, . . . , n + 1 are independent Brownian motions,
τ = inf{t ≥ 0 : Yi (t ) = Yi+1 (t ) for some i},
+
L1− ≡ L n+1
≡ 0 and
L i+ (t )
=
Z
t
1(X i (s) =
Yi (s)) d L i+ (s)
L i− (t )
0
=
Z
t
1(X i (s) = Yi−1 (s)) d L i− (s)
(25)
0
are twice the semimartingale local times at zero of X i − Yi and X i − Yi−1 respectively.
This process can be constructed from the Brownian motions βi and γi by using Skorokhod’s construction to push X i up from Yi−1 and down from Yi . The process is killed when τ is reached, i.e.
when two of the Yi meet.
Warren then goes on to show that the transition densities of this process have a determinantal
−1/2 −x 2 /2t
form similar
e
and
R x to what we have seen in the previous section. Let ϕt (x) = (2πt )
Φt (x) = −∞ ϕt ( y) d y. Let W n,n+1 = {(x, y) ∈ Rn × Rn+1 : x 1 < y1 < x 2 < · · · < yn < x n+1 }.
Define qtn ((x, y), (x 0 , y 0 )) for (x, y), (x 0 , y 0 ) ∈ W n,n+1 and t > 0 to be the determinant of the matrix
–
™
At (x, x 0 ) Bt (x, y 0 )
(26)
Ct ( y, x 0 ) Dt ( y, y 0 )
where
At (x, x 0 ) is an (n + 1) × (n + 1)-matrix where element (i, j) is ϕt (x i0 − x j ),
Bt (x, y 0 ) is an (n + 1) × (n)-matrix where element (i, j) is Φt ( yi0 − x j ) − 1( j ≥ i),
Ct ( y, x 0 ) is an n × (n + 1)-matrix where element (i, j) is ϕt0 (x i0 − y j ) and
Dt ( y, y 0 ) is an n × n-matrix where element (i, j) is ϕt ( yi0 − y j ).
Proposition 4.1 (Prop 2 in [War07]). The process (X , Y ) killed at time τ has transition densities qtn ,
that is
qtn ((x, y), (x, y)) d x 0 d y 0 = P x, y [X (t ) ∈ d x 0 , Y (t ) ∈ d x 0 ; t < τ]
(27)
Warren goes on to condition the Yi not to intersect via so called the Doob h-transform. The transition
densities for the transformed process are given in terms of the those for the killed process by
n,+
qt ((x,
0
0
y), (x , y )) =
hn ( y 0 )
hn ( y)
qtn ((x, y), (x 0 , y 0 )).
(28)
He also shows that you can start all the X i and Yi of the transformed process at the origin by giving
a so called entrance law,
(
)
X
n!
2
t −(n+1) /2 exp −
x i2 /(2t ) hn+1 (x)hn ( y),
(29)
νtn (x, y) :=
Zn+1
i
that is, showing (lemma 4 of [War07]) that this expression satisfies
Z
νtn+s (x 0 , y 0 ) =
n,+
W n,n+1
νsn (x, y)qt
84
((x, y), (x 0 , y 0 )) d x d y.
(30)
It is possible to integrate out the X components in that transition density and entrance law. The
result is transition density
h( y 0 )
n,+
det Dt ( y, y 0 )
(31)
pt ( y, y 0 ) :=
h( y)
and entrance law
µnt ( y) :=
1
Zn
(
2
t −n
/2
exp −
)
X
yi2 /(2t) (hn ( y))2 .
(32)
i
Now comes the interesting part. Let K be the cone of points x = (x 1 , . . . , x m ) where x n =
(x 1n , . . . , x nn ) ∈ Rn subject to the interlacing condition
n
x in ≤ x in−1 ≤ x i+1
(33)
for n = 1, . . . , m and i = 1, . . . , n.
Warren defines a process X(t) taking values in K such that
n,−
X in (t ) = x in + γni (t ) + L i
n,+
(t ) − L i
(t )
n,+
(34)
n,+
where the (γni )i,n are independent Brownian motions and L i and L i are continuous, increasing
n
processes growing only when X in (t ) = X in−1 (t ) and X in (t ) = X i−1
(t ) respectively and the special
n,+
n,−
cases L k and L1 are identically zero for all n.
Think of this as essentially m(m + 1)/2 particles performing independent Brownian motions except
that the n particles in X n can push or block the particles in X n+1 to enforce the interlacing condition
that the whole process should stay in K.
This full process process can be constructed inductively as follows.
n,+
1. The process (X n (t ))t ≥0 has transition densities pt
and entrance law µtn .
n,+
2. The process (X n (t ), X n+1 (t ))t ≥0 has transition densities qt
and entrance law νtn .
3. For n = 2, . . . , m − 1 the process (X n+1 (t))t ≥0 is conditionally independent of
(X 1 (t ), . . . , X n−1 (t ))t ≥0 given (X n (t ))t ≥0 .
n+1,+
4. This implies (by some explicit calculations) that (X n+1 (t ))t ≥0 has transition densities pt
and entrance law µtn+1 .
This argument shows that the following.
Proposition 4.2 (Warren). There exists such a process X(t) started at the origin and it satisfies that
for n = 1, . . . , m − 1, the process (X n , X n+1 ) has entrance law ν tn and transition probabilities q n,+ .
It is this process (X(t ))t ≥0 that is the continuous analog of our discrete process (X (t)) t∈N0 .
85
5
Transition probabilities on two lines
In order to analyse the dynamics described in section 3 we follow Warren’s example and first consider just two lines at a time. What we do in this section is very similar to section 2 of [War07].
Consider the W n+1,n -valued process process (Q n (t)) = (X (t), Y (t)) for t ∈ N0 with components
X (t) = (X 1 (t), . . . , X n+1 (t)) and Y (t) = (Y1 (t), . . . , Yn (t)), satisfying the equations
Yi (t + 1) = Yi (t) + βi (t)
X 1 (t + 1) = X 1 (t) + α1 (t) − 1{X 1 (t) + α1 (t) = Y1 (t + 1) + 1}
X i (t + 1) = X i (t) + αi (t) + 1{X i (t) + αi (t) = Yi−1 (t + 1)}
(35)
− 1{X i (t) + αi (t) = Yi (t + 1) + 1}
X n+1 (t + 1) = X i (t) + αn+1 (t) + 1{X n+1 (t) + αn+1 (t) = Yn (t + 1)}
where αi (t) and βi (t) are i.i.d. coin tosses, s.t. P[αi (t) = 0] = P[αi (t) = 1] = 12 . They evolve until
the stopping time
τ = min t : Yi (t) = Yi+1 (t) for some i ∈ {1, . . . , n − 1} .
(36)
At the time τ the process is killed and remains constant for all time after that. These are very simple
dynamics, each Yi either stays or increases by one independently of all others. The X i do the same
but are sometimes pushed up or blocked by Yi−1 or Yi , respectively, so as to stay in the cone W n,n+1 .
This is the discrete analog of the process (Q(t ))t >0 defined in section 4 of this paper.
Define the forward difference operator and its inverse as
¯ = (−δ0 + δ−1 ) ∗ φ
∆φ
¯ −1 φ(x) =
∆
x−1
X
φ( y).
(37)
(38)
y=−∞
Lemma 5.1. For any f : W n+1,n → R,
X
q0n ((x, y), (x 0 , y 0 )) f (x 0 , y 0 ) = f (x, y).
(39)
(x 0 , y 0 )∈W n+1,n
Proof. Let m = 2n + 1 and z1 = x 1 , z2 = y1 , . . . , zm−1 = yn , zm = x n+1 . Equation (42) in [War07]
states that
¨
«
1{zi ≤ z 0j } i ≥ j
0
det
= 1{z1 ≤ z10 , z2 ≤ z20 , . . . , zm ≤ zm
}
(40)
−1{zi ≤ z 0j } i < j
¯ z )∆z 0 . . . (−∆
¯ z )∆z 0 to both sides of that equality
for z, z 0 ∈ W n . Applying the operator ∆z10 (−∆
2
m−1
m
3
turns the left hand side into q0n ((x, y), (x 0 , y 0 )) and the right hand side into 1{z1 = z10 , z2 = z20 , . . . ,
0
zm = zm
}.
Proposition 5.2. q t , for t = 0, 1, . . . , are the transition probabilities for the process (X , Y ), i.e. for
(x, y), (x 0 , y 0 ) ∈ W n+1,n ,
q tn ((x, y), (x 0 , y 0 )) = P(x, y) [X (t) = x 0 , Y (t) = y 0 ; t < τ]
86
(41)
Proof. Take some test function f : W n+1,n → R. Let
X
F (t, (x, y)) :=
q tn ((x, y), (x 0 , y 0 )) f (x 0 , y 0 )
(42)
(x 0 , y 0 )∈W n+1,n
and
G(t, (x, y)) := E(x, y) [ f (X t , Yt ); t < τ]
(43)
We want of course to prove that F and G are equal and we will do this by showing that they satisfy
the same recursion equation with the same boundary values. By lemma 5.1 we already know that
F (0, ·) ≡ G(0, ·) ≡ f (·).
(44)
The master equation satisfied by G is
G(t + 1, (x, y)) =
1
22n+1
X
G(t, x 1 + a1 , y1 + b1 , x 2 + a2 , . . . , yn + bn , x n+1 + an+1 ). (45)
ai ,bi ∈{0,1}
This formula simply encodes the dynamics that each particle either stays or jumps forward one step.
This needs to be supplemented with some boundary conditions that have to do with the interactions
between particles.
When two of the yi -particles coincide, this corresponds to the event t = τ, which does not contribute
to the expectation in (43). Thus
G(t , . . . , yi−1 = z, x i , yi = z, . . . ) := 0.
(46)
Also, the particle x i cannot jump past yi ,
G(t , . . . , x i = z + 1, yi = z, . . . ) := G(t , . . . , x i = z, yi = z, . . . )
(47)
and x i+1 must not drop below yi + 1,
G(t , . . . , yi = z, x i+1 = z, . . . ) := G(t , . . . , yi = z, x i+1 = z + 1, . . . ).
(48)
G(t + 1, ·) is uniquely determined from G(t, ·) using the recursion equation and boundary values
above. It follows that G is uniquely defined by the recursion equation (45) and the boundary
conditions (44,46,47,48).
Observe that all functions g : Z → R satisfy
1
1
¯
(g(x) + g(x + 1)) = g(x) + ∆g(x).
2
2
(49)
Using this identity many times on (45) shows that
G(t + 1, (x, y)) =
1
1
1
1
1
¯ x )(1 + ∆
¯ y )(1 + ∆
¯ x ) · · · (1 + ∆
¯ y )(1 + ∆
¯ x )G(t, x 1 , y1 , . . . , yn , x n+1 ) (50)
(1 + ∆
1
1
2
n
2
2
2
2
2 n+1
87
which can be rewritten as
¯ t G(t, (x, y)) =
∆
n+1
n
Y
Y
1
1
¯x )
¯ y )−1
(1 + ∆
(1 + ∆
i
i
2
2
i=1
i=1
!
G(t, (x, y)).
(51)
The boundary conditions, equations (46,47,48), can in this notation be rewritten as
G(t, (x, y)) = 0
¯ x G(t, (x, y)) = 0
∆
i
¯ x G(t, (x, y)) = 0
∆
i+1
when yi = yi+1 ,
(52)
when x i = yi and
(53)
when x i+1 = yi .
(54)
Now consider F . The observation (49) gives that φ ∗ ψ = (1 + 21 ∆)ψ. In particular, φ (n+1) ( y − x) =
¯ x )φ (n) ( y − x).
(1 + 1 ∆
2
F (t + 1, (x, y)) =


φ (t+1) (x 10 − x 1 ) ∆−1 φ (t+1) ( y10 − x 1 ) − 1 φ (t+1) (x 20 − x 1 ) . . .


∆φ (t+1) (x 10 − y1 )
φ (t+1) ( y10 − y1 )
∆φ (t+1) (x 20 − y1 ) . . .
 (t+1) 0

 φ
(x 1 − x 2 )
∆−1 φ (t+1) ( y10 − x 2 )
φ (t+1) (x 20 − x 2 ) . . . =


..
.


φ (t) (x 10 − x 1 )
∆−1 φ (t) ( y10 − x 1 ) − 1
φ (t) (x 20 − x 1 )
...


∆φ (t+1) (x 10 − y1 )
φ (t+1) ( y10 − y1 )
∆φ (t+1) (x 20 − y1 ) . . .
1

¯ x )  (t+1) 0
(1 + ∆
φ
(x 1 − x 2 )
∆−1 φ (t+1) ( y10 − x 2 )
φ (t+1) (x 20 − x 2 ) . . . =
2 1 


..
.
n
n+1
Y
Y
1
1
¯
¯ y )F (t, (x, y))
(1 + ∆
= ··· =
(1 + ∆ x i )
i
2
2
i=1
i=1
which shows that F satisfies the same recursion (51) as G.
Now let us consider the boundary values. The probability q tn ((x, y), (x 0 , y 0 )) is zero when yi = yi+1
¯ x q n ((x, y), (x 0 , y 0 )) = 0
because two of its rows are then equal. When yi = x i for some i then ∆
i t
because two rows will be equal when you take the difference operator into the determinant. The
¯ x q n ((x, y), (x 0 , y 0 )) = 0 when yi = x i+1 . Applying this knowledge to
same argument shows that ∆
i+1 t
the sum F , shows that
F (t, (x, y)) = 0
¯
∆ x i F (t, (x, y)) = 0
when yi = yi+1
(55)
when x i = yi
(56)
¯ x F (t, (x, y)) = 0
∆
i+1
when x i+1 = yi
(57)
Since F and G satisfy the same recursion equation with the same boundary values, they must be
equal.
Again, following the example of Warren, we observe that it is possible to condition the processes
never to leave W n,n+1 via a so called Doob h-transform. See for example [KOR02] for details about
88
h-transforms for discrete processes. Recall the definition of Vandermonde determinant in (6). The
h-transform of the process above has transition probabilities
q tn,+ ((x, y), (x 0 , y 0 )) =
hn ( y 0 )
hn ( y)
q tn ((x, y), (x 0 , y 0 )).
(58)
For the sake of notation, call the transformed process (Q n,+ (t)).
The idea now is to stitch together the process X from processes Q k,+ for k = 1, . . . , n − 1, just like
Warren does in the continuous case. For this we need to establish some auxiliary results about Q n
and Q n,+ .
n,+
It is that it is possible to integrate out the x variables from q t .
p tn,+ ( y, y 0 ) :=
Z
0
x 10 ≤ y10 <···≤ yn0 <x n+1
q tn,+ ((x,
0
0
hn ( y 0 )
0
y), (x , y )) d x =
hn ( y)
det[φ (t) ( y 0j − yi )]1≤i, j≤n (59)
n,+
where d x 0 is counting measure on W n+1 . The reader might recognise p t , as a h-transformed
version of the transition probabilities from the Lindström-Gessel-Viennot theorem. Thus this can be
seen as the transition probabilities for a process on W n , where all n particles perform independent
random walks but are conditioned never to intersect, i.e. never to leave W n . We state this as a
proposition.
Fix n > 0 and let x̄ = (1, . . . , n + 1) ∈ Zn+1 and ȳ = (1, . . . , n) ∈ Zn .
Proposition 5.3. Consider the process (Q n,+ (t)) = (X (t), Y (t)) for t ∈ N0 started in (X (0), Y (0)) =
(x̄, ȳ). The process (Y (t)) t∈N0 is governed by p n,+ .
Now for a technical lemma. For x ∈ W n+1 let W n (x) = { y ∈ Zn : x 1 ≤ y1 < · · · ≤ yn < x n+1 } and
for y ∈ W n (x) let
hn ( y)
.
(60)
λn (x, y) = n!
hn+1 (x)
It is not a difficult calculation to show that λn (x, ·) is a probability measure on W n (x). To show this
rewrite hn ( y) as a Vandermonde matrix and perform the summation over all y.
Lemma 5.4.
Z
W n (x)
n,+
n+1,+
λn (x, y)q t ((x, y), (x 0 , y 0 )) d y = p t
(x, x 0 )λn (x 0 , y 0 )
(61)
where d y is counting measure on W n (x), that is integer vectors y that intertwine with x.
n,+
Proof. An elementary calculation given the explicit formula for q t
.
Theorem 5.5. Consider the process (Q n,+ (t)) = (X (t), Y (t)) for t ∈ N0 started in (X (0), Y (0)) =
(x̄, ȳ). The process (X (t)) t∈N0 is governed by p n+1,+ .
Our proof of this is almost identical to the proof of proposition 5 in [War07] and is omitted.
89
6
Transition probabilities for the Aztec Diamond Process
Let us now return to the process (X (t)) t∈N0 that came from the shuffling algorithm. Observe in the
recursions (4), the formulas that define (X n+1 ) t∈N0 contain (X n ) t∈N0 but not (X k ) t∈N0 for k < n.
Thus X n+1 is conditionally independent of (X 1 , . . . , X n−1 ) given (X n ). Also, the dependence of
(X k+1 ) on (X k ) is the same as the dependence of X on Y in Q k and in Q k,+ , see (35). This, together
with theorem 5.5 lends itself to an inductive procedure for constructing the process X .
1. The process (X k (t), t = 0, 1, . . . ) is started at X k (0) = x̄ k and has transition probabilities
governed by p k,+ for k = 1, 2, . . . , k,
2. By proposition 5.3, X k can be considered as the Y component of the process Q k,+ . By the
observation above, the pair of processes (X k (t), X k+1 (t)) has the same distribution as Q k,+
started at (x̄ k , x̄ k+1 ) and are thus governed by transition probabilities q k,+ ,
3. The process X k+1 is conditionally independent of (X 1 , . . . , X k−1 ) given X k .
4. By theorem 5.5, the process X k+1 is governed by transition probabilities p k+1,+ and started at
X k+1 (0) = x̄ k+1 .
This proves theorem 2.1.
7
Asymptotics
p
Let us rescale time by t = N t and space by x i = 21 N t + 12 N x i for i = 0, . . . , n in the above process.
First let us show how to recover the entrance law for Warrens process. Recall that the Aztec diamond
process the discrete process starts at X n (0) = x̄ n and X n+1 (0) = x̄ n+1 .
Lemma 7.1. Fix t and n.
‚ p Œn
N
2
ptn,+ (x̄ n , x) → µnt (x)
(62)
as N → ∞, uniformly for x in compact sets. µnt is defined in (32).
Proof. The expression can be written explicitly as
2
−2n−1
p
2n+1 n,+
N
pt ((x̄ n+1 , x̄ n ), (x,
y)) = 2
−n
p
N
n
hn (x)
hn (x̄ n )
det 2
−t
t
xi − j
n
(63)
i, j=1
which can be evaluated by a formula of Krattenthaler (Theorem 26 of [Kra99]). Applying Stirling’s
approximation to the result shows our lemma.
Lemma 7.2. Fix t and n.
‚ p Œ2n+1
N
2
qtn,+ ((x̄ n , x̄ n+1 ), (x, y)) → ν tn (x, y)
as N → ∞ for x and y in compact sets where ν tn (x, y) is given by (29).
90
(64)
Proof. We can write
p 2n+1 n,+
2−2n−1 N
qt ((x̄ n+1 , x̄ n ), (x, y)) =
2
−2n+1
p
N
2n+1
Z
n,+
x̄ n ∈W (x̄ n+1 )
λn (x̄, ȳ)qt
((x̄, ȳ), (x, y)) d x̄ n (65)
where d x̄ n is counting measure on the space W (x̄ n+1 ) which has only one element. Then we apply
lemma 5.4.
p 2n+1 n+1,+ n+1
= 2−2n+1 N
pt
(x̄
, x)λn (x, y)
(66)
Applying Lemma 7.1 proves this lemma.
Likewise, Warren’s expression for q nt and p nt can be recovered as a scaling limit from our expression
for q tn and p tn . By Stirling’s approximation,
1p
2
Nφ
(t )
−1
(x) → ϕ t (x),
∆
φ
(t )
(x) →
Z
x
ϕ t ( y) d y
(67)
−∞
and
1
4
N ∆φ (t ) (x) →
d
dx
ϕ t (x)
(68)
uniformly on compact sets as N → ∞ where
1
x2
(69)
ϕ t (x) = p
e− 2t .
2πt
p
p
Lemma 7.3. Let x i = 21 N s + 12 N x i , x i0 = 21 N (s + t) + 12 N x i0 , and the same relations for y and y 0 .
p
(
N
2
and
p
)2n+1 qNn t ((x, y), (x 0 , y 0 )) → q nt ((x, y), (x 0 , y 0 ))
N
)n pNn t ((x, y), (x 0 , y 0 )) → p nt ((x, y), (x 0 , y 0 ))
2
uniformly on compact sets as N → ∞.
(
(70)
(71)
Proof. Just insert the limit relations given above for φ and ϕ in the explicit expression for q tn and
p tn .
We now prove the first theorem from section 1.
Proof of Theorem 1.1. Given Corollary 2.2 it is clear that these non-intersecting random walks fit
in as a special case of a model studied in [EK08]. Lemma 4.7 in that article gives the desired
conclusion.
91
For completeness, we here provide our proof of convergence of finite dimensional distributions. Pick
times t 1 < · · · < t m ∈ R and compact sets A1 , . . . , Am ∈ W n . We need to show the following.
Z
Z
d x1 · · ·
lim
N →∞
A1
Am
n,+
n,+
Z
Z
n,+
d x m pt1 (x̄, x 1 )pt2 −t1 (x 1 , x 2 ) · · · ptm −tm−1 (x m−1 , x m ) =
n,+
d x1 · · ·
A1
Am
n,+
n,+
d x m pt1 (x̄, x 1 )pt2 −t1 (x 1 , x 2 ) · · · ptm −tm−1 (x m−1 , x m ) (72)
n
where d x is point measure on W . This is a Riemann sum converging to an integral. By the uniform
convergence of ptn in Lemmas 7.1 and 7.3 we can interchange the order of integration and taking
the limit.
For tightness we shall not here repeat the argument in [EK08]*Lemma 4.7. Suffice it to say that
they show that there exists an α > 2 such that
E[|X N ,n (t2 ) − X N ,n (t1 )|α ] ≤ C|t2 − t1 |α/2
(73)
which according to for example Billingsley [Bil99, Theorem 12.3], is a sufficient condition for tightness.
Finally, one of the main results of this article is the following.
Theorem 7.4. The process (X N ,n (t ), X N ,n+1 (t ))t ≥0 defined in (1) and extended by linear interpolation
to non-integer values of of N t , converges weakly to the process (X n (t ), X n+1 (t ))t ≥0 from Warren’s
process (X(t ))t ≥0 as N → ∞.
Proof. For times t 1 , . . . , t m , and compact sets A1 , . . . , Am ∈ W n,n+1 , we need to study
Z
Z
lim
N →∞
d x1 · · ·
d x m×
Am
A1
n,+
((x̄ n+1 , x̄ n ), (x 1 , y1 ))qt2 −t1 ((x 1 ,
qtn,+
1
n,+
y1 ), (x 2 , y2 )) · · · qtm −tm−1 ((x m−1 , ym−1 ), (x m , x m )) = (74)
where of course d x i d yi is point measure on W n,n+1 . Again this is a Riemann-sum converging to an
integral. By the uniform convergence of qtn in lemmas 7.2 and 7.3, we can interchange the order of
integration and taking the limit. This gives
Z
Z
=
d x 1 d y1 · · ·
A1
d x k d yk ×
Ak
× ν tn (x 1 , y1 )q nt −t ((x 1 , y1 ), (x 2 , y2 )) · · · q nt −t
1
2
1
k
k−1
((x k−1 , yk−1 ), (x k , yk )) (75)
which proves convergence of finite dimensional distributions.
For tightness observe that
E[|(X N ,n (t2 ) − X N ,n (t1 ), X N ,n+1 (t2 ) − X N ,n+1 (t1 ))|α ] ≤
2α/2−1 (E[|(X N ,n (t2 ) − X N ,n (t1 )|α ] + E[|(X N ,n (t2 ) − X N ,n (t1 )|α ]) ≤
2α/2 C|t2 − t1 |α/2 (76)
where the first inequality is due to convexity and the second one is from (73).
92
We feel that given this theorem together with the fact that X n+1 is conditionally independent of
X 1 , . . . , X n−1 given X n , lends a lot of credibility to the following conjecture.
Conjecture. The process (X N (t )t ≥0 defined in (1) converges weakly to Warren’s process (X(t ))t ≥0 as
N → ∞.
The difficulty in proving this is that we cannot write the transition probabilities for the whole process
on a convenient form. Some trick is needed here.
We can say something about the limit at a fixed time.
Proof of Theorem 1.2. Let K m be the cone of points x = (x 1 , . . . , x m ) with x n = (x 1n , . . . , x nn ) ∈ Zn
such that
n+1
x in+1 ≤ x in < x i+1
for i = 1, . . . , n.
(77)
For each x m ∈ W m we will denote by K (x m ) the set of all (x 1 , . . . , x m−1 ) such that
(x 1 , . . . , x m−1 , x m ) ∈ K m . The number of points in K (x m ) is
hn (x m )
Q
card(K (x )) =
.
n<m n!
m
(78)
It follows from the characterisation in section 6 that at a fixed time the distribution of X n−1 (t)
given X n (t) is λn−1 (X n (t), ·). Together with the conditional independence noted in that section
this implies that the distribution of (X 1 (t), . . . , X n−1 (t)) given X n (t) is uniform in K (X n (t)). So the
probability distribution of X (t) for some fixed t ∈ N0 is
mm
t (x)
=
p tm (x̄ m , x m )
χ(x 1 , x 2 ) . . . χ(x m−1 , x m )
card(K (x m ))
(79)
n+1
where χ(x n , x n+1 ) is one if x in+1 ≤ x in < x i+1
for all i = 1, . . . , n and zero otherwise.
n+1
For x m ∈ W m , define K(x m ) as the set of (x 1 , . . . , x m−1 ) where x n ∈ Rn satisfying x in+1 ≤ x in ≤ x i+1
.
m
The n(n − 1)/2-dimensional volume of K(x ) is
hn (x m )
Q
.
vol(K(x )) =
n<m n!
m
(80)
Inserting the rescaling (1) in the expression in (79) and letting N → ∞ using Lemma 7.1 gives
µtm (x)
χ(x 1 , x 2 ) . . . χ(x m−1 , x m )
vol(K(x m ))
(81)
which by [War07, equation (31)] is exactly the probability density for X(t ).
8
Closing Remarks
n,+
Looking at the expression of transition probabilities q t it is natural to ask the question, what
happens if we plug in a different φ than 21 (δ0 + δ1 ) into that determinantal formula? It turns out
that for many other φ this gives a valid transition probability, although we do not fully understand
93
why the Doob h-conditioning still works in that case. It would be interesting to see sufficient and
necessary conditions on φ for this construction to work.
We should mention an article by Dieker and Warren, [DW08]. They study only the top and
bottom particles separately from our model, i.e. in our language (X 11 (t), X 22 (t), . . . , X nn (t)) and
(X 11 (t), X 12 (t), . . . , X 1n (t)) from X (t). They consider both geometric jumps (φ = (1 − q)(δ0 + qδ1 +
q2 δ2 + . . . ), for 0 < q < 1) and Bernoulli jumps (φ = pδ0 + qδ1 where p + q = 1). They write down
transition probabilities but do not do the rescaling to obtain a process in continuous time and space.
Another reference worthy of attention is [Joh07], by Johansson. He considers only geometric jumps
and studies only the top particles (X 11 (t), X 22 (t), . . . , X nn (t)) from our model, with a slight change of
variables that is of no real importance. He not only writes down transition probabilities, but also
recovers the top particles Warren’s process X as the limit of his process properly rescaled.
All the results proved in this article can be generalised to φ = pδ0 + qδ1 where p + q = 1. It is also
not very difficult given the calculations in section 5 to write down transition probabilities for the top
particles and to rescale that to obtain the top particles in Warren’s continuous process, analogous
to Johansson [Joh07] but with Bernoulli jumps. We have not included that calculation here since it
does not add much to our knowledge of these processes, but it is a fact that adds to the plausibility
of the above conjecture.
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